Triangles
What is sine law? What is cosine law? Try to do this without using your notes!
cosine law: c = √(a2 + b2 - 2ab (cos(C)))
sine law: (sin(A)/a) = (sin(B)/b) = (sin(C)/c)
What are the reciprocal trig functions for sin(θ), cos(θ), and tan(θ)?
sin(θ) = 1/csc(θ)
cos(θ) = 1/sec(θ)
tan(θ) = 1/cot(θ)
Find the exact value of sin(165º).
sin(165º) = sin(120º + 45º)
= sin(120º)cos(45º) + cos(120º)sin(45º)
= (√3/2)(√2/2) + (-1/2)(√2/2)
= (√6 - √2) / 4
Draw a graph for the function:
f(x) = -4cos(x)+2
Let's see it!
Can you use cosine law to solve a right triangle? Why or why not?
Yes, you can! The cos of 90º is equal to 0, so it's basically the same as using pythagorean theorem.
What are the 6 trig functions in terms of opposite, adjacent, and hypotenuse? For example:
sin(x) = opposite/hypotenuse
sin(x) = opp/hyp
cos(x) = adj/hyp
tan(x) = opp/adj
csc(x) = hyp/opp
sec(x) = hyp/adj
cot(x) = adj/opp
Daily Double!
Pick another question - in any category - and earn double the points of that question!
Find the exact value of cos(75º).
cos(75º) = cos(30º + 45º)
= cos(30º)cos(45º) − sin(30º)sin(45º)
= (3/√2)(2/√2) − (1/2)(2/√2)
= (√6 - √2) / 4
Daily Double!
Pick another question - in any category - and earn double the points of that question!
Solve the triangle on the board labelled 300.
x ≈ 11.07
If cos(θ) = 1/2, what is cot(θ)?
cot(θ) = 1/√(3)
For the trig identity problem labelled 300, written on the board, find the mistake in the problem AND fix it, so that it gives a correct answer.
Solve the following for an angle in degrees:
3cos(2θ) = -4cos(2θ) + 5
3cos(2θ) = -4cos(2θ) + 5
3cos(2θ) + 4cos(2θ) = 5
7cos(2θ) = 5
cos(2θ) = 5/7
θ = (cos-1(5/7))/2
θ ≈ 22.21
Marisa's, June's, and Daniel's houses form a triangle. The distance between June's and Daniel's houses is 1.2 km. Standing at June's house, the angle formed by looking out to Daniel's house and then to Marisa's house is 63°. Standing at Daniel's house, the angle formed by looking out to June's house and then to Marisa's house is 75°. What is the distance between all of the houses?
Therefore the distances between the homes are:
From Marisa's to Daniel's: 1.6 km
From Marisa's to June's: 1.73 km
From Daniel's to June's: 1.2 km
Solve the triangle on the board labelled 400.
x ≈ 33.18
Using your unit circle for help, find the exact value of cot(5pi/3).
cot(5pi/3) = (x/y)
cot(5pi/3) = (1/2)/(-√(3)/2)
cot(5pi/3) = (1/2)*(2/-√(3))
cot(5pi/3) = -1/√(3)
Solve the following for a final value:
(2cos(t) + 3sin(t))(3cos(t) + 2sin(t)) - 13sin(t)cos(t)
Let c = cos(t), s = sin(t)
= (2c + 3s)(3c + 2s) - 13sc
= 6c2 + 4sc + 9sc + 6s2 - 13sc
= 6c2 + 13sc + 6s2 - 13sc
= 6c2 + 6s2
= 6[cos(t)2 + sin(t)2]
= 6[1]
= 6
Solve the following for angle(s*) in degrees that satisfy the equation:
5cos(θ)=3cos(θ)+2cos(θ)−4cos(θ)
5cos(θ)=3cos(θ)+2cos(θ)−4cos(θ)
5cos(θ)=(3+2−4)cos(θ)
5cos(θ)=cos(θ)
5cos(θ)−cos(θ)=0
4cos(θ)=0
cos(θ)=0
So find all angles in degrees where cos(θ)=0
= 90°, 270°, 450°... etc.
An aircraft is flying between two satellite stations. The distance between the stations is 20 miles. The elevation angle of the aircraft from one station is 15º, and from the other station is 35º. What is the altitude of the aircraft? Make a drawing to help you solve!!!
The altitude of the aircraft is approximately 3.9 miles.
Solve the triangle labelled 500, drawn on the board for all sides and all angles.
alpha = 50º
beta = 100º
gamma = 30º
a = 10
b ≈ 12.9
c ≈ 6.5
Suppose [pi/2 ≤ t ≤ pi], if sec(x) = -10/7, find:
cos(x), csc(x), sin(x), cot(x) and tan(x)
cos(x) = -7/10
csc(x) = 10/√(51)
sin(x) = √(51)/10
cot(x) = -7/√(51)
tan(x) = √(51)/-7
Solve this equation for the angle t, then find all solutions for that angle in the domain [0 ≤ t ≤ 2pi]:
3 sin(t) = (√3)(cos(t))
3 sin(t) = (√3)(cos(t))
(3 sin(t))/(cos(t)) = (√3 (cos (t)))/(cos(t))
(3 sin(t))/(cos(t)) = √3
[3 (sin(t)/cos(t))] = √3
3 tan(t) = √3
tan(t) = √3/3
tan(√3/3)-1 = t
t = 0.524
t + pi = 3.665
Find the exact value of tan(75º). No need to rationalize!
tan(75º) =
(√6 + √2) / (√6 - √2)
Sarah is a tourist visiting the Eiffel Tower in Paris. She stands at a point away from the base of the tower and uses a clinometer to measure the angle of elevation to the top of the tower to be 70º. Moving 200 feet closer, the angle of elevation to the top of the tower is 80º. How tall is the Eiffel Tower?
The height of the Eiffel Tower is approximately 1064.62 feet.